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August  2015, 9(3): 375-390. doi: 10.3934/amc.2015.9.375

Some new results on cross correlation of $p$-ary $m$-sequence and its decimated sequence

Wenbing Chen 1, , Jinquan Luo 2, , Yuansheng Tang 3, and  Quanquan Liu 3,

1. 

School of Mathemetics & Computation Science, Anqing Normal University, Anqing, Anhui 246133, China
2. 

Department of Mathematic and Statistics, Centtal China Normal University, Wuhan, Hubei 430079, China
3. 

Department of Mathematic Sciences, Yangzhou University, Yangzhou, Jiangsu 225002, China, China

Received  September 2014 Published  July 2015

Let $p$ be an odd prime, $n=2m$, and $n/\gcd(k,n)$ be odd. In this paper, we study the cross correlation between a $p$-ary $m$-sequence $(s_{t})$ of period $p^{n}-1$ and its decimated sequence $(s_{dt})$ where $d$ satisfies $d(p^k+1)\equiv p^m+1 \pmod {p^n-1}$. Our results show that the cross-correlation function is six-valued and the distribution of the cross correlation is also completely determined.
Keywords: $p$-ary sequence, exponential sum, Cross correlation, quadratic form..
Mathematics Subject Classification: Primary: 94A55; Secondary:11B5.
Citation: Wenbing Chen, Jinquan Luo, Yuansheng Tang, Quanquan Liu. Some new results on cross correlation of $p$-ary $m$-sequence and its decimated sequence. Advances in Mathematics of Communications, 2015, 9 (3) : 375-390. doi: 10.3934/amc.2015.9.375
References:
[1]

A. W. Bluher, On $x^{q+1}+ax+b$,, Finite Fields Appl., 10 (2004), 285.  doi: 10.1016/j.ffa.2003.08.004.  Google Scholar

[2]

W. Chen, J. Luo and Y. Tang, Exponential sums from half quadratic forms and its applications,, in Proc. ISIT'14, (2014), 3145.   Google Scholar

[3]

S. T. Choi, J. S. No and H. Chung, On the cross-correlation of a ternary m-sequence of period $3^{4k+2}-1$ and its decimated sequence by $(3^{2k+1}+1)^{2}$ over 8,, in Proc. ISIT'10, (2010), 1268.   Google Scholar

[4]

S. T. Choi, J. S. No and H. Chung, On the cross-correlation of a $p$-ary m-sequence of period $p^{2m}-1$ and its decimated sequence by $\frac{(p^m+1)^2}{2(p+1)}$,, IEEE Trans. Inf. Theory, 58 (2012), 1873.  doi: 10.1109/TIT.2011.2177573.  Google Scholar

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H. Dobbertin, P. Felke and T. Helleseth, Niho type cross correlation functions via Dickson polynomials and Kloosterman sums,, IEEE Trans. Inf. Theory, 52 (2006), 613.  doi: 10.1109/TIT.2005.862094.  Google Scholar

[6]

T. Helleseth, Some results about the cross-correlation function between two maximal-linear sequence,, Discrete Math., 16 (1976), 209.   Google Scholar

[7]

R. Lidl and H. Niederreiter, Finite Fields,, Addison-Wesley, (1983).   Google Scholar

[8]

J. Luo and K. Feng, On the weight distribution of two classes of cyclic codes,, IEEE Trans. Inf. Theory, 54 (2008), 5332.  doi: 10.1109/TIT.2008.2006424.  Google Scholar

[9]

J. Luo and K. Feng, Cyclic codes and sequences from generalized Coulter-Matthews function,, IEEE Trans. Inf. Theory, 54 (2008), 5345.  doi: 10.1109/TIT.2008.2006394.  Google Scholar

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J. Luo, T. Helleseth and A. Kholosha, Two nonbinary sequences with six-valued cross correlation,, in Proc. IWSDA'11, (2011), 44.   Google Scholar

[11]

J, Luo, Y. Tang and H. Wang, Exponential sums, cyclic codes and sequences: the odd characteristic Kasami case,, preprint, ().   Google Scholar

[12]

G. J. Ness, T. Helleseth and A. Kholosha, On the correlation distribution of the Coulter-Matthews decimation,, IEEE Trans. Inf. Theory, 52 (2006), 2241.  doi: 10.1109/TIT.2006.872857.  Google Scholar

[13]

E. Y. Seo, Y. S. Kim, J. S. No and D. J. Shin, Cross-correlation distribution of p-ary m-sequence of period $p^{4k}-1$ and its decimated sequences by $(\frac{p^{2k}+1}{2})^{2}$,, IEEE Trans. Inf. Theory, 54 (2008), 3140.  doi: 10.1109/TIT.2008.924694.  Google Scholar

[14]

Y. Sun, Z. Wang, H. Li and T. Yan, The cross-correlation distribution of a $p$-ary m-sequence of period $p^{2k}-1$ and its decimated sequence by $\frac{(p^k+1)^2}{2(p^e+1)}$,, Adv. Math. Commun., 7 (2013), 409.  doi: 10.3934/amc.2013.7.409.  Google Scholar

[15]

Y. Xia, C. Li, X. Zeng and T. Helleseth, Some results on cross-correlation distribution between a $p$-ary $m$-sequence and its decimated sequences,, IEEE Trans. Inf. Theory, 60 (2014), 7368.  doi: 10.1109/TIT.2014.2350775.  Google Scholar

show all references

References:
[1]

A. W. Bluher, On $x^{q+1}+ax+b$,, Finite Fields Appl., 10 (2004), 285.  doi: 10.1016/j.ffa.2003.08.004.  Google Scholar

[2]

W. Chen, J. Luo and Y. Tang, Exponential sums from half quadratic forms and its applications,, in Proc. ISIT'14, (2014), 3145.   Google Scholar

[3]

S. T. Choi, J. S. No and H. Chung, On the cross-correlation of a ternary m-sequence of period $3^{4k+2}-1$ and its decimated sequence by $(3^{2k+1}+1)^{2}$ over 8,, in Proc. ISIT'10, (2010), 1268.   Google Scholar

[4]

S. T. Choi, J. S. No and H. Chung, On the cross-correlation of a $p$-ary m-sequence of period $p^{2m}-1$ and its decimated sequence by $\frac{(p^m+1)^2}{2(p+1)}$,, IEEE Trans. Inf. Theory, 58 (2012), 1873.  doi: 10.1109/TIT.2011.2177573.  Google Scholar

[5]

H. Dobbertin, P. Felke and T. Helleseth, Niho type cross correlation functions via Dickson polynomials and Kloosterman sums,, IEEE Trans. Inf. Theory, 52 (2006), 613.  doi: 10.1109/TIT.2005.862094.  Google Scholar

[6]

T. Helleseth, Some results about the cross-correlation function between two maximal-linear sequence,, Discrete Math., 16 (1976), 209.   Google Scholar

[7]

R. Lidl and H. Niederreiter, Finite Fields,, Addison-Wesley, (1983).   Google Scholar

[8]

J. Luo and K. Feng, On the weight distribution of two classes of cyclic codes,, IEEE Trans. Inf. Theory, 54 (2008), 5332.  doi: 10.1109/TIT.2008.2006424.  Google Scholar

[9]

J. Luo and K. Feng, Cyclic codes and sequences from generalized Coulter-Matthews function,, IEEE Trans. Inf. Theory, 54 (2008), 5345.  doi: 10.1109/TIT.2008.2006394.  Google Scholar

[10]

J. Luo, T. Helleseth and A. Kholosha, Two nonbinary sequences with six-valued cross correlation,, in Proc. IWSDA'11, (2011), 44.   Google Scholar

[11]

J, Luo, Y. Tang and H. Wang, Exponential sums, cyclic codes and sequences: the odd characteristic Kasami case,, preprint, ().   Google Scholar

[12]

G. J. Ness, T. Helleseth and A. Kholosha, On the correlation distribution of the Coulter-Matthews decimation,, IEEE Trans. Inf. Theory, 52 (2006), 2241.  doi: 10.1109/TIT.2006.872857.  Google Scholar

[13]

E. Y. Seo, Y. S. Kim, J. S. No and D. J. Shin, Cross-correlation distribution of p-ary m-sequence of period $p^{4k}-1$ and its decimated sequences by $(\frac{p^{2k}+1}{2})^{2}$,, IEEE Trans. Inf. Theory, 54 (2008), 3140.  doi: 10.1109/TIT.2008.924694.  Google Scholar

[14]

Y. Sun, Z. Wang, H. Li and T. Yan, The cross-correlation distribution of a $p$-ary m-sequence of period $p^{2k}-1$ and its decimated sequence by $\frac{(p^k+1)^2}{2(p^e+1)}$,, Adv. Math. Commun., 7 (2013), 409.  doi: 10.3934/amc.2013.7.409.  Google Scholar

[15]

Y. Xia, C. Li, X. Zeng and T. Helleseth, Some results on cross-correlation distribution between a $p$-ary $m$-sequence and its decimated sequences,, IEEE Trans. Inf. Theory, 60 (2014), 7368.  doi: 10.1109/TIT.2014.2350775.  Google Scholar

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